In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime p. More precisely, it states that for all integer polynomials
, either: where deg f is the degree of f. This can be stated with congruence classes as follows: for all polynomials
with p prime, either: If p is not prime, then there can potentially be more than deg f(x) solutions.
be an integer polynomial, and write g ∈ (Z/pZ)[x] the polynomial obtained by taking its coefficients mod p. Then, for all integers x,
Furthermore, by the basic rules of modular arithmetic,
Both versions of the theorem (over Z and over Z/pZ) are thus equivalent.
We prove the second version by induction on the degree, in the case where the coefficients of f are not all null.
If deg f = 0 then f has no roots and the statement is true.
If deg f ≥ 1 without roots then the statement is also trivially true.
The fact that Z/pZ is a field allows to apply the division algorithm to f and the polynomial x − k (of degree 1), which yields the existence of a polynomial
(of degree lower than that of f) and of a constant
(of degree lower than 1) such that
[1] The other roots of f are then roots of g as well, which by the induction property are at most deg g ≤ deg f − 1 in number.
Let p(X) be a polynomial over an integral domain R with degree n > 0.
Then the polynomial equation p(x) = 0 has at most n = deg(p(X)) roots in R.[2]